C₃×#2199 #72914

order	384
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

	subgroup	G/H
center, socle	C₂×C₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂³×C₃
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃ ◁ C₂×C₃×D₄ ◁ G

270 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,19);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(3,10)(4,12)(6,13)(9,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5490);